The application of digital techniques to the testing and analysis of the analog properties of signals has allowed the development of equipment able to perform, with precision, a wide variety of tests and analysis that are difficult or impossible to perform with purely analog techniques. To cite just one example, a spectrum analyzer whose internal operation is of a digital nature can, for a relatively modest expenditure of additional computational resources, provide phase information in addition to amplitude, and can then provide information about a signal's modulation. In this connection with this particular example, IQ modulation is a very general form of modulation that is specified through amplitude/phase pairs, and both Amplitude Modulation and Frequency Modulation can be described as special cases of IQ modulation, not to mention the various specialized types of IQ modulation that are used as various standards of digital modulation. To achieve the same degree of functionality (not to mention reliability, small size and low power consumption) with the purely analog techniques of several decades ago is altogether impractical.
The starting point for the digital treatment of analog properties of a waveform is the application of the waveform of interest to an ADC (Analog to Digital Converter). This is a circuit that accepts an analog signal as its input and produces an output that is a numeric code of n-many bits taken at some sample rate, usually regular in that the samples are equally spaced in time. The n-many bits are sometimes regular binary, sometimes a Gray code (adjacent codes in a monotonic sequence differ by a transition in only one bit position), with larger values for n producing higher resolution. Six, eight and ten are common values of n; sometimes n might be smaller and sometimes it is larger. There is a generally accepted tradeoff concerning n and how fast the ADC can operate: above some limits, faster operation is facilitated by a smaller value of n, and for a given n, there is a limit on how fast the ADC can go.
Before proceeding, we should dispose of one potentially confusing aspect related to terminology: this business of ‘digital’ and ‘analog’ signals. We view this as an issue related to how signals are used to represent information. At the lowest level of abstraction any tangible signal has an analog nature in the sense that it is something measurable and persists for some period of time. In short, it exists as “something real,” and there might be a little more of it one time and a little less another. An ‘analog signal’ is a signal the very stuff of which is used to convey the information of interest. That is, some continuously variable property of the signal IS the information (e.g., at what rate is the signal changing, what are its levels, etc.) A ‘digital signal’ is essentially a code represented by some analog property of that signal, or most frequently, collection of such signals. The analog properties of the signal are only used to convey the digital code, and within very broad limits minor variations in those analog properties will not change the code at all. It is noted that we are ignoring here the entire notion of ‘digital’ logic, and of its AND, OR and NOR gates, etc. Insofar as the analog behavior of those ‘digital’ signals comports with the rules for correctly expressing encoded content, we are content to call them ‘digital’ signals and otherwise ignore their analog nature. It is a convenient fiction related to the manner in which signals are intended to convey information. In this sense, analog signals are their own representation, as it were, of their information content, and various circuits do various things to their properties. This is the realm of ‘traditional’ analog electronics. The sense in which we will use the notion of ‘digital signals’ is in the discrete numerical representation of some analog property that probably came from some ‘signal’ of interest. Much competent ‘traditional electronics’ can be needed to correctly get that numerical representation, but once it is on hand the way it is manipulated is arithmetic (as is data by a computer), not ‘electronically’ (e.g., push-pull amplification in an amplifier), although electronics will undoubtedly provide the means to do it (just as it does within a computer).
In any event, the fundamental philosophy of the digital treatment of a signal's analog properties is to digitize the analog signal with an ADC having suitable resolution, store the numeric (digital) values in a memory and process the data either in real time or via post processing, depending upon the application. ‘Processing’ usually means computation using the stored numeric values.
The selection of the number n for the resolution provided by the ADC is not the only major parameter of ADC performance (of which there are several . . . ) that is of interest. Of particular interest is the sustained rate at which the ADC can track or respond to a changed analog input and provide a new digital output. The experience of early real-time DSOs (Digital Sampling Oscilloscopes) is illustrative. It was readily appreciated that one had to sample a signal ‘faster than it changed’ in order to create a faithful trace whose waveform might be arbitrary, and more importantly, occurred perhaps just once. (Repetitive waveforms, on the other hand, lend themselves to being sampled at a low rate while letting the location of the samples “drift across” the waveform to build up an “equivalent time” acquisition record over many passes that could not be obtained as a single unified “real time’ acquisition record by sampling one signal instance densely—which is to say, faster.) A somewhat simplistic early view of what the sampling did was that it produced a trace composed of just dots-at-amplitudes that, when they were close enough together along the time axis, provided a useable replica (a trace) of the real waveform. Clearly, frequency components within the original signal and whose periods were less than the time between samples would not contribute correctly to the replica represented by the trace. It was soon appreciated, however, that their presence in the original at the time it is applied to the ADC can actually do a fair amount of damage to the replica, and it was seen as necessary to apply a low pass filter to the signal before sampling it.
The analysis of Fourier that allowed any repetitive waveform to be decomposed and then expressed as a summation of sinusoids of particular frequencies, amplitudes and phases was neither unknown nor unappreciated, but the hardware (low cost memory and fast arithmetic processors) and practical algorithms needed to take advantage of this knowledge, took a few years for commercial development and refinement to occur. Computer programs that analyzed complex waveforms existed from the earliest days of computing. This algorithmic capability wasn't yet in an oscilloscope, or any other test equipment. As that further development and refinement occurred, the theorems of Nyquist and Shannon became the gate keepers of what data should be converted from analog to digital to perform what has come to be called DSP (for Digital Signal Processing). Entire DSP processing mechanisms were eventually reduced to integrated circuitry, and the modern DSO was afoot (as was an architectural revolution for many other types of test equipment). With ADCs and DSPs it was possible to sample ‘at about’ twice the rate of the highest frequency component of interest and let the algorithms ‘find all the dots.’ That is, it is even possible to create a nice smooth ‘stretched-out’ segment of displayed trace that appears to have all its pixels in wonderful harmony, when in reality only samples corresponding to ‘about’ every nth pixel have actually been taken. Such is the power of Nyquist and Fourier as combined in modern DSP techniques.
So, we are not surprised that, in certain applications such as real time DSOs and other items of high frequency test equipment such as spectrum analyzers, there is an urge for the ADC (a digitizer) to ‘go fast.’ In the usual terminology, which is the one we shall use below, the ADC has a certain sampling rate or frequency FS at which it operates. While an ADC might sometimes be operated at less that its maximum FS, it does have such a maximum, and for a great many applications the highest frequency component FMAX that a signal will be allowed to approach (but not reach) will be FMAX=FS/2. An anti-aliasing filter will be used to enforce that requirement before the signal is applied to the ADC. Failure to do so can create great mischief. For example, consider a one megahertz sine wave that is (too slowly) sampled every one micro-second, and let's say these numbers are exact. Every sample is going to have the same value, depending upon the phase of the sampling regime relative to the signal. The sine wave has been converted into a straight line. It is most unlikely that such is what was intended. On the other hand, if we observe the Nyquist requirement and either sample at less than every half micro-second, or keep the original sampling rate and limit the signal to less than half a megahertz, then we get samples that faithfully preserve the original information and support an analysis or even a reconstruction of the original waveform.
There is yet another aspect of high speed ADC operation that needs appreciation. When the speed of ADC operation (i.e., its rate of sustained conversion) gets above that at which available memory (e.g., RAM, or Random Access Memory, which is to say, read-write memory) can store that data at a sustained rate, some interesting arrangements emerge, such as the interleaving of memory into banks, placing custom memory cells onto the same IC (Integrated Circuit) substrate as the ADC, etc. In short, things get complicated at the implementation level, even though at the simplified block diagram level it appears simply that an analog signal is digitized by an ADC and the digital words stored in a memory. So, in many high performance items of electronic test equipment the owner is somewhat dismayed to discover that she cannot increase the size of its memory with the same ease that has become usual and customary in the realm of computers, whether of the Personal or Workstation variety. It is not just an issue of does the processor provide the extra addressing; the authors of the embedded system within the test equipment knew ahead of time that such simply was not going to happen, because of a specialized non-standard manner in which that memory was closely coupled to the operation of the ADC in order to obtain the desired speed of operation.
Thus, when someone contemplating a new instrument design opines that “We need an ADC that goes twice (or three) times as fast as what is available . . . ” it might not be simply a matter of finding somebody who knows how to provide it. A whole bunch of other stuff may also need to be bored out to permit the increase in performance. On the one hand, this might merely be a recipe for an expensive top-of-the-line item of test equipment (assuming all the relevant problems can be solved), or, on the other, it might be a boundary above which performance does not rise, regardless of the expense incurred (we simply can't make an ADC that runs faster, or if we did, we can't store the data . . . ).
Sometimes, when the answer to the question is unpleasant or inconvenient, there may be a way to get a more satisfactory answer if we discover a somewhat different question to ask. So, if we are ordinarily faced with digitizing signals that occupy a frequency band BIN of DC to FMAX, and the Nyquist requirement tells us to sample signals in BIN at a sampling rate FS>2FMAX, and it so happens that we do not have at our disposal an ADC that operates any faster than FMAX, we might ask what advantages accrue to us that we might exploit if we could guarantee that a signal FIN that we were interested in at any one time always stayed in the region DC to FMAX/2 or in FMAX/2 to FMAX. That is, it stays nicely in either the bottom half or the upper half of BIN. Suppose it does. The customary response is to divide BIN into two bands B1 and B2. We then treat digitized material for B1 as adequately Nyquist compliant in something we will call the “First Nyquist region” (DC to FS/2), provided it truly was always within B1, while digitized material for B2 is likewise adequately Nyquist compliant for the “Second Nyquist region” (FS/2<FIN<FS), provided that it was always within B2. The “always within” requirement is enforced by switching anti-aliasing bandpass or low pass filters into the signal path ahead of the ADC.
We immediately recognize this ‘First Nyquist region’ as what was related to the earlier discussion of sampling at FS>2FMAX. We are lead to expect that we can correctly deal with signals (and/or their components) from DC up to (but not including) FMAX in frequency, provided frequencies at or above FMAX are absent. But what is this business of a Second Nyquist region? And might there be a Third Nyquist region, and so on? Just what are these additional Nyquist regions?
To explain this notion of Nyquist regions, let's suppose a sampling rate of once every micro-second, and consider a signal frequency of 250 kHz (0.25 MHz) and see what happens. First, the period of the signal to be sampled is 4 μs, and upon reflection it is unlikely that all four one micro-second samples are going to have the same amplitude: there is no group of n-many equally spaced points anywhere along the time axis and within the period of a sine wave that all have the same amplitude, for n greater than two. If there were two samples, then yes, there would be lots ways two samples could have the same amplitude, but once there are more than two, that cannot happen. This is classic Nyquist in action. (It might be objected that there are indeed other periodic waveforms that can produce (n>2)-many samples of equal value. They won't be sinusoids. Let's avoid this worry by agreeing that we have the Fourier decomposition at hand, and pick a sampling rate that will have n>2 for the highest frequency component of interest in that decomposition. What works for that sine wave will automatically work for all the others, and the issue is eliminated, since all the other sinusoids are of lower frequency and n>>2 for them.)
Okay, so what else can we say about the four samples we get? Well, for one thing, we can fit a 250 kHz sine wave to them. After all, that is where they came from. But, and this is what is of interest to us here, so also can a 1.25 MHz sine wave of the same amplitude as before be fitted to those points. Why is this? It is because the period between samples divides the period of the 250 kHz signal into some fractional portion, which in this easy case is one quarter cycle. (That in this example ¼ is a ‘nice’ number is not important—it could be 2/7 or some truly nasty number. It won't matter.) So we have this correspondence: The first sample gets (say) the location a first quarter of the way along the first cycle of the waveform, the second sample gets a location the second quarter of the way, the third sample gets the third quarter, the fourth sample the fourth quarter, and the fifth sample gets the location at the first quarter along the second cycle, and so on. The sequence of locations within a cycle is ¼, 2/4, ¾, 4/4 followed in the next cycle by the locations ¼, 2/4, et seq.
When we substitute 1.25 MHz for 0.25 MHz, the new signal to be sampled has one and a quarter cycles for every quarter cycle of the old one. The new sequence of samples is: the first sample gets the location at one and a quarter cycles, the second sample get the location at two and half cycles, the third sample gets the location at three and three quarters cycles, the fourth at four and four fourths, and the fifth sample gets the location at five and five fourths cycles. But this is the same sequence of locations within some cycle: ¼, 2/4, ¾, 4/4, ¼, et seq. It is true that these locations are not ones along the same cycle, but they are for the same respective locations along consecutive cycles. If no one told us where they came from, we would be none the wiser and might well believe that these samples were obtained from the 250 kHz signal. We would be wrong and not know it. We readily see that this mischief (called aliasing) repeats itself with the addition of every integral multiple of the sampling frequency to the signal frequency. So, if we substitute 2.25 MHz for the 0.25 MHz, the new signal has two and a quarter cycles for every quarter cycle of the original one. You can see it coming: the sequence of sampled locations is again equivalent to every quarter of a cycle, albeit distributed amongst successive (but not consecutive) cycles.
To make this sort of thing easy to talk about, we say that the sampling regime, which in this case is to take a sample every one micro-second, has Nyquist regions. The First Nyquist region of the sampling regime in this set of examples is DC to one half the sampling frequency, and is sampling according to the most stringent set of Nyquist rules. The Second Nyquist region is sampling at the same rate as for the First Nyquist region, but upon signals FS/2 higher (i.e., FS/2 to FS) and that, by rights, ought to (but can't, or won't) have twice the sampling frequency, or 2FS. The Third Nyquist region is sampling (again) at the same rate as for the First Nyquist region, but upon signals 2FS/2 higher (i.e., FS to 3FS/2). To generalize, then, the Nth Nyquist region for FS is to sample at FS but upon signals that are in the frequency range from (N−1)FS/2 to NFS/2.
It is clear that aliasing is potentially a very serious problem, and the only sure cure is to avoid it. Hence, anti-aliasing filters and band limited operation to within a selected band patrolled by the filters. The idea is that the ambiguity is removed, since the signal that was digitized first went through a known anti-aliasing filter: the digitized values must represent a signal that lies within the pass band of that filter. If need be, a correct reconstruction or interpretation of the digitized result can be formed by ‘adding’ an appropriate frequency offset depending upon the Nyquist region represented by the anti-aliasing filter that was in use when the digitized samples were taken, and inverting the frequency when the Nyquist region number is even.
This business of consecutive Nyquist regions enforced by anti-aliasing filters works, but there is an unpleasant complication: Two adjacent (in terms of their pass bands) practical anti-aliasing filters do not form a true partition of BIN into B1 and B2. If one had filters whose skirts were truly vertical (each a so-called ‘brick wall’ filter) one might be tempted to try exactly that. As is well known, however, such ‘brick wall’ filters are not realizable in practice. Practical filters have real skirts that are not vertical, and filters that have really steep skirts tend to have severe disadvantages in other departments. If we overlap the skirts to ensure at least adjoining passbands, then there will be signals that are within that overlap of the skirts and thus appear within the passbands of both filters. This creates ambiguity about which Nyquist region the signal is to belong—there is in principle no way to tell, since the anti-aliasing requirement has not actually been met for such a signal! Oops! Thus it is that anti-aliasing filters have non-overlapping disjoint guard bands that are to enforce a suitable separation between the passbands, in order to eliminate the possibility of such ambiguity. The unpleasant complication is that signals whose spectral location is within the disjoint region between/within the guard bands cannot be utilized. It is a recognition that there are no adjacent passbands for anti-aliasing filters.
It would seem that we are compelled to waste some of Mother Nature's spectrum. But perhaps this is another case where we should ask a different question in hopes of finding a more agreeable answer. Upon some reflection, we appreciate that the notion of adjacent (which for all practical purposes means overlapping) anti-aliasing filters is propelled by the notion of adjacent Nyquist regions. So, what is it that makes two Nyquist regions adjacent? Perhaps there is sufficient difference between the real meaning of “adjacent or overlapping passbands” for mere filters and the genuine “Nyquist requirement” for sampling band limited analog signals, that we can with grace and elegance recover access to that otherwise ‘wasted’ spectrum, and wrest it back from the clutches of those evil guard bands. Hmmm. But how to do it?